Table Of ContentSpringer Series on
ATOMIC, OPTICAL, AND PLASMA PHYSICS 40
Springer Series on
ATOMIC, OPTICAL, AND PLASMA PHYSICS
The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive
manner theory and experiment in the entire field of atoms and molecules and their
interaction with electromagnetic radiation. Books in the series provide a rich source of
new ideas and techniques with wide applications in fields such as chemistry, materials
science, astrophysics, surface science, plasma technology, advanced optics, aeronomy,
and engineering. Laser physics is a particular connecting theme that has provided much
of the continuing impetus for new developments in the field. The purpose of the series
is to cover the gap between standard undergraduate textbooks and the research
literature with emphasis on the fundamental ideas, methods, techniques, and results in
the field.
27 Quantum Squeezing
By P.D. Drumond and Z. Ficek
28 Atom, Molecule, and Cluster Beams I
Basic Theory, Production and Detection of Thermal Energy Beams
By H. Pauly
29 Polarization, Alignment and Orientation in Atomic Collisions
By N. Andersen and K. Bartschat
30 Physics of Solid-State Laser Physics
By R.C. Powell (Published in the former Series on Atomic, Molecular, and Optical Physics)
31 Plasma Kinetics in Atmospheric Gases
By M. Capitelli, C.M. Ferreira, B.F. Gordiets, A.I. Osipov
32 Atom, Molecule, and Cluster Beams II
Cluster Beams, Fast and Slow Beams, Accessory Equipment and Applications
By H. Pauly
33 Atom Optics
By P. Meystre
34 Laser Physics at Relativistic Intensities
By A.V. Borovsky, A.L. Galkin, O.B. Shiryaev, T. Auguste
35 Many-Particle Quantum Dynamics in Atomic and Molecular Fragmentation
Editors: J. Ullrich and V.P. Shevelko
36 Atom Tunneling Phenomena in Physics, Chemistry and Biology
Editor: T. Miyazaki
37 Charged Particle Traps
Physics and Techniques of Charged Particle Field Confinement
By V.N. Gheorghe, F.G. Major, G. Werth
38 Plasma Physics and Controlled Nuclear Fusion
By K. Miyamoto
39 Plasma-Material Interaction in Controlled Fusion
By D. Naujoks
40 Relativistic Quantum Theory of Atoms and Molecules
By I.P. Grant
Vols. 1–26 of the former Springer Series on Atoms and Plasmas are listed at the end of the book
I.P. Grant
Relativistic Quantum Theory
of Atoms and Molecules
Theory and Computation
I.P.Grant
Mathematical Institute
Oxford University
Oxford,UK
[email protected]
Library ofCongress Control Number:2006926733
ISBN-10:0-387-34671-6 Printed on acid-free paper.
ISBN-13:978-0387-34671-7
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Printed in the United States ofAmerica. (MVY)
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For my wife, Beryl, my sons, Paul and David, and my
grandchildren Jacob, Georgia, Joshua, and Imogen
Preface
Relativistic atomic structure theory has been widely used for many years
for predicting atomic energy levels, transition rates, collision cross sections
and many other properties. The results have been important for fundamental
physics experiments as well as providing data for space science, controlled
fusion, and industrial applications. The development of a relativistic theory
of the electronic structure and properties of molecules and condensed matter
is more recent because it is technically more complex and needs more elabo-
rate mathematical machinery than nonrelativistic theory. The first attempts
in the early 1980s at relativistic molecular electronic structure calculations
based on Dirac’s Hamiltonian were unsuccessful and were mostly abandoned
in favour of semi-relativistic approximations for the bulk of applications to
quantum chemistry. Now that the dust has settled, there is a need for a book
on rigorous foundations that sets out workable and economical methods of
fully relativistic calculations for atoms, molecules and clusters that can be
used by both physicists and chemists.
Part I of the book, which is nonspecialist, aims to equip the reader to rec-
ognizethequalitativesignatureofrelativisticeffectsintheelectronicstructure
of atoms and molecules. Part II deals with the theoretical foundations of the
field.Theformofrelativisticwaveequationsisdeterminedbythegeometryof
Minkowski space-time and the structure of the Lorentz and Poincar´e groups.
Quantumelectrodynamicsdescribesthephysicsoftheinteractionofelectrona
andelectromagneticfields;theequationsaretoocomplicatedtosolveexactly,
but we can write down systematic approximation schemes that have proved
very effective for modelling electronic structure.
This provides the foundation for practical applications to atomic and
molecular physics in Part III of the book. The electrostatic potential near
each atomic nucleus is almost spherical, so that the nonrelativistic electron
wavefunctionhasacharacteristiccentralfield characterinthatregion,permit-
tingfactorizationintoradialandangularparts.Relativisticeffectsonelectron
dynamics are most marked in the strong electric field near the nucleus; the
consequentialcouplingofthefourcomponentsofDiraccentralfieldspinorsin
VIII Preface
thatregionthereforecharacterizesrelativisticatomicwavefunctions.Thelong-
range electron-electron interaction propagates relativistic effects right across
the atom or molecule, so that it is essential to treat the whole many-electron
system relativistically in order to get reliable results.
The coupled radial components of Dirac atomic spinors can be approxi-
mated numerically with finite differences, finite elements, or by expansion in
analytic functions. Only the last approach is viable for polyatomic systems.
Justasnonrelativisticmolecularwavefunctionsareoftenapproximatedaslin-
earcombinationsofGaussianfunctions(GTF),sorelativisticmolecularwave-
functionscanbeconstructedfromG-spinors–four-componentgeneralizations
of GTF having relativistic central field character. These incorporate all the
internal relations between the components to satisfy the boundary conditions
at the nuclei. Nonrelativistic GTF and G-spinors both represent the electron
distribution between atoms and the resultant molecular bonding in much the
sameway.Thefailuresofthe1980swereduetotheincorrectassumptionthat
the four components of a Dirac spinor can be considered independent and
that variational calculations made with this assumption would reproduce the
internal structure of orbital spinors with sufficient accuracy. Successful four
component calculations need a spinor basis set.
Thebodyofthebookpresentsthetechnologyneededforpracticalcalcula-
tions;theappendicescontainsupplementarymaterial.Chapters6to8setout
the mathematical machinery of relativistic electronic structure calculations
on atoms and ions, illustrated with output from a version of the GRASP
computer package. Chapter 9, centred on the DARC relativistic R-matrix
package,discussesmainlyelectron-atom/ionscattering,photo-excitation,and
photo-ionization. The construction and application of the relativistic molecu-
lar structure code BERTHA are described in Chapters 10 and 11. Appendix
A lists frequently used mathematical formulae, whilst Appendix B presents
mathematical background material on linear operators in Hilbert space, Lie
groups and Lie algebras, angular momentum theory (including diagram tech-
niques), and various aspects of numerical approximation including the theory
ofvariationalmethodsforDiracoperatorsanditerativesolutionof(MC)DHF
equations.
The book is primarily intended as a resource for research physicists and
chemists, experimental or theoretical, who recognize that using the available
relativistic electronic structure packages as “black boxes” is not always wise.
These readers need to understand the physical and theoretical background in
order to appreciate what can be done with existing codes and. what is just
as important, what cannot be done. I have tried to include enough detail so
that this book can be used by graduate students starting work in the field as
well as by experts. Some material, especially Chapter 1, should be accessible
to undergraduates. More difficult sections marked with an asterisk (∗), can
be skipped on a first reading.
When the late Charles Coulson first suggested that I write a book on rel-
ativistic atomic structure theory more than 30 years ago, I had no inkling
Preface IX
how much the subject would develop nor how long the writing would take.
I regret I have not been able to find more space for some important topics
such as relativistic many-body theory and the calculation of radiative cor-
rections. The relativistic atomic structure package GRASP owes its present
form to many collaborators: in particular Nicholas Pyper, Steven Rose, Neil
Beatham, Bruce McKenzie, Jiro Hata, Ken Dyall, Patrick Norrington, and
Farid Parpia. Charlotte Froese Fischer who, with Farid Parpia and others,
has developed GRASP92 for modern multi-processor computers in order to
study complex electronic structures in heavy atoms, continues to introduce
further innovations, and her influence can be seen on both Chapters 7 and 8.
I am also grateful to Stephan Fritzsche, whose RATIP procedures take rela-
tivistic atomic calculations in a new direction, and to illuminating correspon-
dence with Steven Manson on photoionization theory and RRPA methods.
The DARC code for relativistic R-matrix calculations, Chapter 9, much of
which is based on GRASP, was mainly developed by Patrick Norrington and
Wasantha Wijesundera. The BERTHA relativistic molecular structure code,
Chapters 10 and 11, involved a close collaboration with Harry Quiney and
Haakon Skaane. Harry Quiney’s Oxford D. Phil. thesis laid the foundations
of the relativistic basis set method in 1987 which were implemented in the
DHF atomic code that preceded BERTHA. This book reflects the innumer-
ablediscussionsonallaspectsofrelativisticatomicandmoleculartheorythat
we have had over the past 20 years. I am most grateful for his careful reading
of much of the draft, which has resulted in many improvements. All errors
and omissions are, of course, my responsibility alone.
Oxford, Ian Grant
August 2006
Contents
Part I Relativity in atomic and molecular physics
1 Relativity in atomic and molecular physics ................ 3
1.1 Elementary ideas ........................................ 3
1.2 The one-electron atom ................................... 7
1.2.1 Classical Kepler orbits ............................. 7
1.2.2 The Bohr atom ................................... 8
1.2.3 X-ray spectra and Moseley’s Law.................... 9
1.2.4 Transition to quantum mechanics.................... 9
1.2.5 Sommerfeld’s relativistic orbits and Dirac’s wave
equation ......................................... 10
1.2.6 Dirac and Schro¨dinger charge distributions ........... 13
1.2.7 The Dirac hydrogenic spectrum at high Z ............ 17
1.3 Many-electron atoms..................................... 19
1.3.1 Central field models of the atom .................... 20
1.3.2 Closed and open shells ............................. 21
1.3.3 Mean field potentials .............................. 23
1.3.4 Comparison of Hartree-Fock and Dirac-Hartree-Fock
models for ground states ........................... 24
1.3.5 The mechanism of shell filling....................... 35
1.3.6 Other approaches ................................. 38
1.4 Applications to atomic physics ............................ 40
1.4.1 X-ray spectra ..................................... 41
1.4.2 Applications to astrophysics and plasma physics....... 43
1.4.3 Modelling atomic processes in plasmas ............... 44
1.5 Relativistic molecular structure ........................... 45
1.5.1 Relativistic interpretations of chemical anomalies...... 46
1.5.2 Relativistic effective core potentials and other
approximations ................................... 48
1.5.3 Dirac four-component methods for molecules.......... 51
XII Contents
1.5.4 Parity violation and hyperfine interactions............ 52
1.5.5 High-precision spectroscopy of small molecules
containing light elements ........................... 54
References..................................................... 54
Part II Foundations
2 Relativistic wave equations for free particles............... 63
2.1 The special theory of relativity............................ 63
2.2 The Lorentz group....................................... 66
2.2.1 ∗Spinor representation of Lorentz transformations..... 67
2.2.2 ∗Infinitesimal Lorentz transformations and their
generators........................................ 69
2.2.3 ∗Representations of the Lorentz group............... 70
2.3 The Poincar´e group...................................... 73
2.3.1 ∗Representations of the Poincar´e group .............. 75
2.3.2 ∗Space and time reflections......................... 77
2.4 The Klein-Gordon equation............................... 81
2.5 The Dirac equation ...................................... 86
2.5.1 γ-Matrices and covariant form of Dirac’s equation ..... 87
2.5.2 ∗Lagrangian formulation of Dirac’s equation.......... 88
2.5.3 Foldy canonical form and the Foldy-Wouthuysen
transformation .................................... 90
2.5.4 ∗Position operators in Dirac theory ................. 92
2.5.5 Dirac particles in electromagnetic fields .............. 94
2.5.6 ∗Negative energy states............................ 95
2.6 Maxwell’s equations ..................................... 96
2.6.1 Covariant form of Maxwell’s equations ............... 96
2.6.2 ∗Lagrangian formulation........................... 99
2.6.3 Gauge invariance..................................100
2.6.4 ∗Motion of a test charge ...........................102
2.7 ∗Symmetries and local conservation laws...................103
2.8 ∗Global conservation laws................................107
2.9 ∗Green’s functions ......................................108
2.9.1 Nonrelativistic Green’s functions ....................109
2.9.2 Klein-Gordon operator .............................112
2.9.3 Maxwell’s equations: the zero-mass case ..............116
2.9.4 Free-particle Dirac equation ........................118
References.....................................................119
